I know this is a REALLY old post, but I wanted to clear up some historical points...
The first name of the art was indeed "Hwa Soo Do" meaning the way of the flowering hand. This was the founder, Hwang Kee's first attempt at connecting to old Korean culture, and honoring the Hwa Rang Do warriors. It's also said that the flowering hand is a reference to the blend of hard and soft as well as the use of both closed and open hands in the system.
He failed to get his system off the ground twice because it had too much of a Chinese flavor for the newly independent Korean country which was occupied by Japan for 36 years.
He had learned Okinawan/Japanese Hyung from books he found in stations along the Chosun railway. In fact, many older Koreans remember that when you speak of the Moo Duk Kwan, you speak of the Railway Dojangs. After the difficulties of opening his school, it's noted that he did go to train with Lee Won Kuk of the Chung Do Kwan, who was teaching Kong Soo Do (Empty Hand Way), or basically Karate-Do.
Hwang Kee learned these hyung and applied his own methods for developing power, even rearranging the movements in some cases, or kicking higher than their Japanese counterparts. Passai Hyung, for example, includes a high side kick, while Bassaidai, the Japanese Kata, included a sidekick to the leg. (Pal Che, the Southern Shaolin, original set may have included something altogether different.)
He re-opened his doors as "Hwa Soo (Tang Soo) Do - Moo Duk Kwan." There's a picture floating around the internet that shows the Hanja in parentheses...
Since his personal education did have more Chinese influence, it was only fair to use Tang Soo Do (Chinese Hand Way) which was also a regularly used term during the time.
He produced his first set of Yu Dan Ja in 1947, but by this point, most people called the style "Tang Soo Do Moo Duk Kwan" and the Hwa Soo part was simply dropped.
In 1957, Hwang Kee was presented with the Moo Yei Dobo Tong Ji, in which the founder identified the Kwon Bup (Fist Method) section... and found the term for all methods of defense without weapons, including grappling, known as Soo Bahk. It is said that Soo Bahk in its ancient form was then split into Tae Kyon (the striking side) and Ssireum (the grappling side).
The rest is history, but the unclear part is when the names were adopted, and who took what name upon leaving the founder:
The founder officially filed the name "Dae Han Soo Bahk Do Hoi, Moo Duk Kwan" (Korean Soo Bahk Do Association, the School of Martial Virtue) on June 30, 1960. Rather, it filed was earlier, but that's when the filing was approved.
A lot of people LOVED the name Tang Soo Do because of the amount of time they'd spent instructing and being students of it, so to change to "Soo Bahk Do" was difficult. This is why all throughout the 60's, 70's, 80's, and early 90's, in the U.S., Tang Soo Do is what was prevalent.
The U.S. Tang Soo Do Moo Duk Kwan Federation, Inc. was chartered on June 28, 1975... they adopted the name finally in 1994 to reflect the founder's original wishes to honor the ancient Korean martial heritage.
So Soo Bahk Do is the name of the art studied by students who were officially recognized by the World Moo Duk Kwan organization, and most people who left took the name Tang Soo Do.